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# Tropical Geometry and Kapranov's Theorem Open Access

## Descriptions

Other title
Subject/Keyword

Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Cuevas Pineda, Guillermo Javier
Supervisor and department
Kuttler, Jochen (Mathematical and Statistical Sciences)
Examining committee member and department
Troitsky, Vladimir (Mathematical and Statistical Sciences)
Chernousov, Vladimir (Mathematical and Statistical Sciences)
Gille, Sthephan (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2013-09-30T13:30:39Z
2013-11
Degree
Master of Science
Degree level
Master's
Abstract
The tropical variety of a function $$f=\sum c_ux^u\in K[x_1,\cdots,x_n]$$ is the set of points $$r\in\mathbb{R}^n$$ where the minimum of $$\val(c_u)+\langle r,u\rangle$$ is attained at least twice. We define the initial form of a function with respect to a vector $$r$$, $$\IN_r(f)$$, to be the sum of the monomials of $$f$$ with least valuation. Our main focus is on the points $$w$$ where the initial form is not a monomial. Such points can be lifted to points $$z\in (K^*)^n$$ satisfying $$f(z)=0$$ and $$\val(z)=w$$. This result is known as Kapranov's Theorem and has been previously proved using geometric properties, field extensions, and different definitions of what an initial form is (including the one given above). In this thesis we prove Kapranov's theorem using the aforementioned definition of an initial form and its algebraic properties. Keywords: Tropical Geometry; Initial forms; Kapranov's Theorem.
Language
English
DOI
doi:10.7939/R3ZC7S31B
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
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